First Order Optics Jordan Jur 1. INTRODUCTION First order optics are the principles and equations which describe the geometrical imaging of any optical system. The foundations of first order optics are derived from the concept of central projection, collinear transformation and the camera obscura. These foundations will be used to demonstrate the fundamental parameters of an optical system such as image position, size and orientation. These fundamental parameters will be calculated from the different cardinal points of an optical system using both Newtonian and Gaussian imaging equations. To extrapolate upon these imaging principles, the derivation of Snell’s law and the law of reflection will be covered. 2. FUNDAMENTAL CONCEPTS Central Projection The geometry of first order optics is extrapolated from the concept of central projection. The central projection concept states that a given object located in the object plane which is projected through the “projection point” will form an image of the object in the image plane. The central projection concept is an application of the mathematical concept of projection. Projection is defined as a mapping of a set into a subset. In relation to the central projection concept, the set encompasses the points and/or lines which define the object in object space, the mapping function would be the projection point and the subset encompasses the image points and/or lines which define the object in image space. Collinear transformation is a type of projection which has applications to optics, as well. Figure 1. Central Projection Collinear Transformation A collinear transformation is a one to one mapping between two spaces. In the world of optics, the spaces are referred to as object space and image space. All of the points within object space pass through the same projection point which maps them to points within the image space. The projection function of a collinear system states that points map to points, lines map to lines and spaces map to spaces. The points, lines and spaces in object space all have corresponding and unique points, lines and spaces in image space. The notion of these corresponding and unique elements describes the basis of conjugate elements. Next, the camera obscura is described to illustrate the idea of image formation with rays. This is a necessary concept in order to build upon the mathematical principles and equations which govern the geometric behavior of the object and image equations. Further details of these imaging equations and their response to small perturbations can be described by linear shift invariant systems theory. The Camera Obscura The camera obscura is the simplest form of an imaging system. It consists of a black box with a pinhole in one side. Light from an object placed outside of the box will propagate through the pinhole and form an inverted image on the wall. Here light can be thought of as a ray which travels from left to right to form the image. Notice that the geometry and concept of the camera obscura is an adaptation of the central projection concept. An important detail of this imaging system lies in the pinhole diameter. Figure 2. Camera Obscura As the pinhole diameter increases so does the image brightness. This is because a larger aperture will enable more light to pass through the optical system. Though, if the aperture is too large then the image will become blurry. This, of course, describes the preliminary concept of resolution and the point spread function (PSF). Conversely, as the pinhole diameter decreases, thereby improving image resolution and PSF, so does the image brightness. Before expanding upon the complexities of resolution and PSF, the geometric relationship between the object and image distances must be understood. The distance from the object to the pinhole is known as the object distance while the distance from the pinhole to the image is known as the image distance. The idea of object and image distances leads to the mathematical concepts of imaging as independently explain by Sir Isaac Newton and Karl Gauss. With the concepts of central projection, collinear transformation and the camera obscura laid out, a mathematical and physical geometric basis can now be constructed to help derive and illustrate, respectively, the first order properties of an optical system. 3. IMAGING EQUATIONS The imaging equations can be derived by tracing the chief and marginal rays of an optical system. The marginal ray is an on-axis ray which travels from the center of an object to the edge of the stop and to the center of the image. The chief ray is an off-axis ray which travels from the edge of the object through the center of the stop to the edge of the image. In other words, the marginal and chief rays define the extrema of rays which define an image to the first order. That is to say, all other rays which can be traced through an optical system lie between the marginal (image center) and chief (image edge) rays. The image location is known as the focal plane and is our first introduction into the cardinal points. Figure 3. Marginal and Chief Rays The cardinal points define points of angular and spatial significance within an optical system. Any optical system can be decomposed in terms of the six cardinal planes. The six cardinal planes are the front and rear focal planes, nodal planes and the principle planes. The focal planes are the planes at which all rays parallel to the optical axis in object space will come to focus in image space. The same is true for rays traveling from image space towards object space. The nodal planes are the planes of unit angular magnification. A ray which passes through the front nodal point will pass through the rear nodal point at the same angle with respect to the optical axis. The principle planes are the planes of unit lateral magnification. A ray which passes through the front principal point will pass through the rear principal point at the same vertical distance from the optical axis. The front and rear cardinal points are conjugate to one another. For reference, the symbolic interpretation of the Newtonian and Gaussian imaging systems is explain in table 1. Symbol Meaning F Front Focal Plane P Front Principal Plane z Object Distance fF Front Focal Distance h Object Height n Object Space Refractive Index F’ Rear Focal Plane P’ Rear Principal Plane z’ Image Distance f'R Rear Focal Distance h' Image Height n' Image Space Refractive Index m Magnification fE Table 1. Symbol Meaning Newtonian Imaging Equations The imaging equations were first derived by Sir Isaac Newton in 1666 using similar triangles. In relation to the conjugate object and image planes, the Newtonian equations are referenced to the focal planes. Figure 4. Newtonian Imaging Equations Figure 5. Newtonian Imaging Geometry Gaussian Imaging Equations Gauss later derived similar imaging equations where the conjugate object and image planes are referenced to the principal planes. Both sets of imaging equations assume that the lenses are thin and therefore the principal and nodal planes lie at the center of the lens. Figure 6. Gaussian Imaging Equations Figure 7. Gaussian Imaging Geometry 4. OPTICS LAWS Law of Reflection The simplest form of ray propagating can be understood by the law of reflection. The law of reflection states that for a ray incident upon a flat surface at an angle of α with respect to the surface normal reflect away from the flat surface at an angle of –α with respect to the surface normal. Figure 8 provides a visualization of the reflection which takes place at a flat surface. Figure 8. Law of Reflection Law of Refraction: Snell’s Law In any optical system, light travels through different types of media. As light travels from one media to another the direction of propagation will alter depending upon the incident angle and the properties of both media. The simplest model which describes the change in direction is known as Snell’s Law. Snell’s Law states that a ray in media 1 which is incident upon a surface at an angle of α with respect to surface normal will alter its direction into media 2 at angle β with respect to the surface normal. This altering of direction is known as refraction. The equation which mathematically describes Snell’s Law is shown in equation 1. 푛1 sin 훼 = 푛2 sin 훽 (1) Where n1 and n2 are the indices of refraction for the incident and transmitted surface. One can image that any time a ray of light interacts with different media that Snell’s Law can be applied to determine the change in the direction of propagation. Figure 9 provides a visualization of the refraction which takes place at an interface between two different media. Figure 9. Law of Refraction When the transmitted medium has a higher refractive index than the incident medium the ray will bend towards the surface normal. When the transmitted medium has a lower refractive index than the incident medium the ray will bend away from the surface normal. The principles of reflection and refraction can be applied to optical systems to understand how object and image motion are affected. 5. OBJECT AND IMAGE MOTION Object and image motion is described in relation to a three dimensional Cartesian coordinate system. This coordinate system has six effective degrees of freedom: x, y & z motion and θX, θY, & θZ rotation, as described in figure 10. There exist reflective and refractive optical elements which will alter the ray propagation through an optical system and thus effect the object and image motion. Common reflective elements include mirrors and prisms and common refractive elements include plane parallel plates (PPP) and lenses.